nLab triple category

Idea

A triple category is an n-fold category for $n=3$. That is, it is an internal category in double categories. This concept can be considered at various levels of strictness. However, there is not yet a definition in the literature of a fully weak (i.e. pseudo) triple category.

Examples

• strict triple category?
• intercategory
• The double category of algebras can be enhanced to a triple category
• Just as any category gives rise to a double category of commutative squares, any double category gives rise to a triple category of commutative “cubes” by taking the transversal morphisms to be the same as the vertical morphisms.